The farthest distance a satellite signal can directly reach is the length of the tangent to the curve of Earth's surface. If the angle formed by the tangent satellite signals is 104°, what is the measure of the intercepted arc (x) on Earth?
Hint: If minor arc AC = x then remember that major arc ABC = 360 - x. You will need to substitute into your equation provided in the theorem.
Show all work.
(This is 10th grade math, please don't make it advanced, shorten the steps.)
First, we need to use the fact that the angle formed by the tangent satellite signals is 104 degrees to find the central angle (θ) of the intercepted arc on Earth:
θ = 180 - 104 = 76 degrees
Next, we use the formula for calculating the length of an arc:
length of arc = (θ/360) x 2πr
where r is the radius of the Earth.
Since we want to find the length of the intercepted arc (x), we can set up the following equation:
x = (θ/360) x 2πr
We know that the radius of the Earth is approximately 6,371 kilometers, so we can substitute that value in and simplify:
x = (76/360) x 2π(6,371)
x = (0.2111) x 40,075.16
x ≈ 8,470 kilometers
Therefore, the measure of the intercepted arc (x) on Earth is approximately 8,470 kilometers.
To find the measure of the intercepted arc (x) on Earth, we need to apply the tangent theorem. According to the theorem, if the angle formed by the tangent satellite signals is 104°, the measure of the intercepted arc (x) is equal to twice the measure of the angle.
Therefore, x = 2 * 104°
x = 208°